CIRP 12
Viktor P. Astakhov
Similarity Numbers in Metal Cutting
Testing and Modeling
What seems to be a problem?
Viktor P. Astakhov CIRP12 2009
2
Studies of Metal Cutting
Introduction
1. Analytical Studies
2. Numerical Studies
3. Experimental Studies
Problems with Experimental Studies
1. High cost
2. Long time
3. Particularity of the obtained results
Viktor P. Astakhov CIRP12 2009
3
"In theory, there is no difference between theory and
practice. But, in practice, there is."
Jan L.A. van de Snepscheut
Viktor P. Astakhov CIRP12 2009
4
Example of the machining system (drilling)
Introduction
HIERARCHY OF THE COMPONENTS OF THE MACHINING SYSTEM (DRILLING)
TOOL
MACHINE
Tool holder
Stability of spindle
rotation under axial
load and drilling
torque
Shank
Tip
Quality of carbide
Geometric and
kinematic
accuracy of the feed
motion
FIXTURE
Accuracy and
repeatability of
clamping
Static and Dynamic
rigidity
WORKPIECE
OPERATOR
Repeatability of
work material
properties
Clear work
instruction
(ISO 9001)
Accuracy of the
datums
COOLANT SYSTEM
Training to act under
normal and
abnormal
situations
Surface finish
Periphery grind
Static and dynamic
rigidity
Location and shape
of the coolant holes
Coolant rotary union
Detailed Control Plan
(the coolant type and its maintenance schedule)
Concentration
and composition
Flow rate
(Pressure)
MAINTENANCE
Clear work
instruction
(ISO 9001)
Point design and
geometry
Coating
Alignment "spindlestarting bushing"
pH (for water
soluble)
Temperature
(in and out)
Schedule and
apparatus
Repeatability of
point geometry
in re-sharpenings
Maintenance control
plan and schedule
Total and active EP ed.
(for straignt oils)
Filtration
Decision making
logic
Proper handling
and storage
CONTROL SYSTEM
Parameters to
display
Parameters
to control
Transducers and
signal conditioning
Viktor P. Astakhov CIRP12 2009
5
Introduction
Similarity Theory
The similarity theory offers a better way to obtain a sound
mathematical model of the complicated processes taking
place in a complex technical system. Today it is largely
used in the area of thermodynamic, fluid flow etc. This
theory combines various information and knowledge
about a complex process under study. Its basic principle
is separation of a group of similar phenomena from a
great class of phenomena by a general low. In such a
context, similarity can be geometrical, physical etc.
Viktor P. Astakhov CIRP12 2009
6
Introduction
Similarity Theory in Metal Cutting
At the present stage, however, the similarity theory is not
yet developed in metal cutting studies. Rather, a number
of useful similarity criteria (numbers) are developed that
can be used in modeling of the metal cutting process.
The objective of this presentation is to discuss three
most important similarity numbers as the chip
compression ratio, the Péclet and the Poletica
numbers.
Viktor P. Astakhov CIRP12 2009
7
Chip Compression Ratio (CCR)
t1
Chip compression
ratio
Tool-Chip Interface
v - Cutting speed
t
Cutting
direction
t2
v
 t 2  v
1
1
v1 - Chip velocity
Chip
lc
Tool
Workpiece
f
Tool
Cutting feed
Viktor P. Astakhov CIRP12 2009
8
Significance of CCR
The elementary work spent over plastic deformation of a unit volume of
the work material calculates as
dA  Au   i i  1.74 K  ln   1.15ln   2 K  ln  
CCR
n
n 1
K is the stress at ε=1, n is the strain-hardening coefficient.
Knowing CCR, one not only assure the similarity of the deformation process
but also calculate the power spent on the plastic deformation of the layer
being removed and power spent due to friction at the tool-chip interface.
These two are major contributors to the total power required by the cutting
system
Ppd =
K 1.15lnζ 
n +1
n+1
vA w
Viktor P. Astakhov CIRP12 2009
9
Significance of CCR
Aluminum 2024T6
Steel 52100
Energy of plastic deformation, 67%
CCR
Energy of plastic deformation, 63%
Rake energy, 22%
Cohesive energy, 6%
Flank energy, 9%
Cohesive energy, 7%
Rake energy, 20%
Flank energy, 6%
CCR is the simplest yet most important and most objective characteristic
of the cutting process
Viktor P. Astakhov CIRP12 2009
10
Pèclet number
Pe number
Definition for metal cutting
vt1
Pe 
ww
ww =
kw
(c p  ) w
where v is the velocity of a moving heat source (the cutting speed) (m/s), w w is the
thermal diffusivity of the work material (m2/s), kw is the thermoconductivity of the
work material, (J/(m·s·oC)), (cp ·)w is the volume specific heat of work material,
(J/(m3·oC)).
The Péclet number is a similarity number, which characterizes the relative influence of the
cutting regime (vt1) with respect to the thermal properties of the workpiece material (ww). If
Pe>10 then the heat source (the cutting tool) moves over the workpiece faster than the
velocity of thermal wave propagation in the work material so the thermal energy generated in
cutting due to the plastic deformation of the work material and due to friction at the tool-chip
interface does not affect the work material ahead of the tool. If Pe<10 then the thermal energy
due to the plastic deformation and due to friction makes its strong contribution to the process
of plastic deformation during cutting as its affect the mechanical properties of the work
Viktor P. Astakhov CIRP12 2009
11
material.
Practical use in testing
Pe number
0.125mm/rev
2.4
2.4
0.200mm/rev
0.390mm/rev
2.2
2.2
0.280mm/rev
0.75mm/rev
2.0
2.0
0.500mm/rev
1.8
1.8
0
1.0
2.0
3.0
v (m/s)
0
70
140
280
Pe
Influence of the cutting speed on CCR for
Generalization of the experimental
different speeds. Work material – steel AISI
data using the Péclet number
1030, tool material – carbide P20, rake angle
γn = 10o, cutting edge angle κr = 60o, depth of
cut dw = 2mm
Viktor P. Astakhov CIRP12 2009
12
Practical use in testing
f=0.1mm/rev
3.0

Pe number
0.2mm/rev
:
0.3mm/rev
2.0
0
1

(a)
2
v(m/s)
+20°
-10°
+10°
-20°
0°
2.8
2.0
3.0
1.2
70
2.0
0
70
(b)
140
Pe
CCR vs. (a) the cutting speed for different feeds and (b)
Pe criterion. Work material – tool steel H13, tool material
– carbide M10, rake angle γn =−10o, cutting edge angle
κr = 60o, depth of cut dw = 2mm
140
210
Pe
CCR vs. Pe criterion for different rake angles. Work
material – steel AISI 1045, tool material – carbide
P20, cutting edge angle κr = 60o, depth of cut dw =
2mm
Viktor P. Astakhov CIRP12 2009
13
Po number
Poletica number (Po-criterion)
In metal cutting, the tool–chip contact length known as the length of the tool–chip
interface determines major tribological conditions at this interface as temperatures,
stresses, tool wear, etc. Moreover, all the energy required by the cutting system for
chip removal passes through this interface. Therefore, it is of great interest to find out
a way to asses this length.
To deal with the problem, the Poletica criterion (Po-criterion) is introduced as the
ratio of the contact length, lc to the uncut chip thickness, t1
lc
Po 
t1
Viktor P. Astakhov CIRP12 2009
14
Practical use in testing
Po
Po
10
Po number
8
8
6
6
4
4
f = 0.07 0.15
 =-10°
 = 0°
 = 10°
 = 20°
2
0
1
2
3
4
5
0.26
0.34
(mm/rev)
HB110
HB200
HB320
2
0

Influence of chip compression ratio on
Po-criterion in machining steel AISI
E9310, tool material P20 (79%WC,
15%TiC, 6%Co), cutting feed f = 0.07
− 0.43mm/rev and cutting edge angle
κr = 70o
1
2
3
4
5
6

Influence of chip compression ratio on Pocriterion in machining beryllium copper
UNSC17000 of different hardnesses. Tool
material – M30 (92%WC, 8%Co)
Viktor P. Astakhov CIRP12 2009
15
Practical use in testing
Po
Copper, UNSC17000 cutter, = 25°
Copper, Ti Grade 1cutter, = 25°
Steel O7, annealed, = 10°
Copper, HSS M35 cutter, = 10°
Beryllium copper, HB110, = 10°
Beryllium copper, HB200, = 10°
Beryllium copper, HB320, = 10°
Armco iron, = 10°
Introduction
40
30
20
10
Steel E9310,M30 cutter, = -10°,0°,10°,20°
0
1
5
9
13
17
t
Influence of chip compression ratio on Po-criterion in machining various
work materials using different tool materials and tool rake angles
Viktor P. Astakhov CIRP12 2009
16
Viktor P. Astakhov CIRP12 2009
17
Other important numbers
A number
One of the most important is the A-criterion. Since it first derived and studied by
Silin, it may be referred as Silin criterion. It calculates as
A
t1b1T c c
Fp
and characterizes the part of the thermal energy (heat) absorbed by the chip
relative to the whole amount of heat generated in the deformation zone. In this
equation t1, b1T are the uncut chip thickness and the true chip width, respectively,
m, cρ is the volumetric heat capacity of the work material, J/(m3 oK); θc is the
cutting temperature, oC; Fp is the power components of the force, N.
Viktor P. Astakhov CIRP12 2009
18
The D-criterion which calculates as
D, E, F numbers
tt
D
b1T
and characterizes the uncut chip cross-section.
The E-criterion or relative sharpness of the cutting edge which calculates
E
1
t1
and characterizes the influence of the cutting edge radius ρ1 (m) with respect to the
uncut chip thickness t1 (m).
The F-criterion which calculates as
kt
F
 n tn
kw
characterizes influence of the tool geometry with respect to the thermal conductivities
of tool and work materials. In tis equation, kt and kw are thermal conductivities of tool
and work materials, J/(m s oC), respectively, βn is the normal tool wedge angle; εtn is
the acute angle in the reference plane between the major (side) and minor cutting
edges.
Viktor P. Astakhov CIRP12 2009
19
Machinability test
Machinability
m4
n3 Pe
A m m m
F 5D 6E 7
where constants n3, m4 – m7 are to be determined experimentally using a suitable
design of experiment techniques
In experimental studies of machinability when a specific tool (tool material, tool
holder etc) and workpiece (dimensions and work material) were selected for test,
it often sufficient at the first stage of the study to consider the following
relationship
Pe  n1 A
m1
or
 t1b1T c  c
vt1
 n1 
 Fp
ww




m1
where n1 and m1 are constants to be determines experimentally.
Viktor P. Astakhov CIRP12 2009
20
Machinability test
Machinability
The latter equation can be re-written for the optimum cutting speed vo (the speed
that corresponds to the optimal cutting temperature o) as
n1 ww
vo 
t1
 t1b1T c  o

Fp




m1
To determine constants n1 and m1, the power components of the force,
Fp and the cutting temperature c are measured simultaneously. If the
test results are plotted on a double logarithmic A versus Pe diagram
(the same module along both axes) as shown, then n1 = Pe when A = 1
and m1 = tan 1. For data shown in the figure, the machinability
equations becomes
9.1ww
vo 
t1
 t1b1T c  o

Fp




2.3
Experimental determination of the constants of Eq. (15): (a) work material - stainless
steel AISI 303, tool material: carbide P01 (66%WC30%TiC4%Co), tool geometry: γn =
12o, αn = 10o, κr = 45o, κr1 = 25o, rn = 1 mm, similarity numbers: F = 1.48, D = 0.01260.1500, E = 0.06 - 0.76.
Viktor P. Astakhov CIRP12 2009
21
Machinability
Machinability test
The foregoing analysis leads to a new approach to machinability determination using the following
procedure. Five - seven different cutting feeds should be selected for the study. The depth of cut should
be kept the same for all tests. The number of tests corresponds to that of the selected cutting feeds. In
each test, the cutting speed is varied and the cutting force and cutting temperature are measured. As
shown in the figure.
Workpiece material: nickel-based high alloy
(0.08%C1%Cr56%Ni1%Co1%Al), tool material: carbide M30
(92%WC8%Co), tool geometry: γn = 12o, αn = 12o, κr = 45o, κr1
= 45o, rn = 1 mm, cutting regime: dw = 1 mm, f, mm/rev, =10.074, 2- 0.11, 3- 0.15, 4- 0.25, 5- 0.30, 6- 0.34, 7- 0.39
Viktor P. Astakhov CIRP12 2009
22
Machinability test
on
The optimum cutting speed is defined for each feed as that corresponding to the
minimum stabilized value of the cutting force. Plotting the results on a double
logarithmic the true uncut chip thickness versus cutting speed (same module along
both axes), one can obtain a t1 - v curves as shown. This t1 - v curve may be
considered linear within a certain range of the uncut chip thickness. The equation
for this linear proportion of the curve is written
v (m/s)
vo  n t
m2
21
0.40
0.30
0.20
In which constants n2=0.034 and m2 = 0.81.
SIMPLE, physically-grounded, straightforward
test
0.10
0.08
0.03
0.06
0.1
0.2
0.3
Viktor P. Astakhov CIRP12 2009
23
Conclusions
To narrow the gap between the metal cutting theory and practice, a sound
similarity approach should be developed to utilize the full power of the
similarity theory.
In the author’s opinion, the basic set of the relevant similarity numbers
should be developed in metal cutting and the three basic theorem of
similarity should be used to determine the necessary and sufficient
conditions of similarity of cutting process. The three first similarity numbers
discussed here, namely, CCR, the Péclet and Poletica criteria are of a
great help in metal cutting studies.
Viktor P. Astakhov CIRP12 2009
24
THANK YOU
The happy end
Direction of spending
Viktor P. Astakhov CIRP12 2009
25